If \(\theta _i\)  are exponential\((\lambda )\) , then
\(a_N^{-1}=1\)  (see 1.7.2. in [1]}). Consider \(S_{\tau , N, T}\)  and pick \(u_{S, \tau , N, T}\)  as in eq. (REF ).
Then since
\(F_{Gu, {\lambda }}^{-1}\left(1- \dfrac{1}{N\tau } \right) & = \dfrac{\log (N\tau )}{\lambda }~,\\F_{Gu, {\lambda }}^{-1}\left(1- \dfrac{1}{N\tau } \dfrac{\log (T)}{\log (T)+1} \right) & = \dfrac{\log (N\tau ) + \log \left(\frac{\log (T)+1}{\log (T)} \right)}{\lambda }~,\) 
